Besides standard Lagrange interpolation, i.e., interpolation of target functions from scattered point evaluations, positive definite kernel functions are well-suited for the solution of more general reconstruction problems. This is due to the intrinsic structure of the underlying reproducing kernel Hilbert space (RKHS). In fact, kernel-based interpolation has been applied to the reconstruction of bivariate functions from scattered Radon samples in computerized tomography (cf. Iske, 2018) and, moreover, to the numerical solution of elliptic PDEs (cf. Wenzel et al., 2022). As shown in various previous contributions, numerical algorithms and theoretical results from kernel-based Lagrange interpolation can be transferred to more general interpolation problems. In particular, greedy point selection methods were studied in (Wenzel et al., 2022), for the special case of Sobolev kernels. In this paper, we aim to develop and analyze more general kernel-based interpolation methods, for less restrictive settings. To this end, we first provide convergence results for generalized interpolation under minimalistic assumptions on both the selected kernel and the target function. Finally, we prove convergence of popular greedy data selection algorithms for totally bounded sets of functionals. Supporting numerical results are provided for illustration.

翻译：除了标准的拉格朗日插值（即通过散点函数值进行目标函数插值）外，正定核函数非常适用于解决更一般的重构问题。这归因于其底层再生核希尔伯特空间的内在结构。事实上，核基插值已应用于计算机断层扫描中基于散射拉东样本的双变量函数重构（参见Iske, 2018），并进一步应用于椭圆型偏微分方程的数值求解（参见Wenzel等人, 2022）。如先前多项研究所表明，核基拉格朗日插值的数值算法与理论结果可推广至更一般的插值问题。特别地，（Wenzel等人, 2022）针对Sobolev核的特殊情形研究了贪婪点选择方法。本文旨在针对限制更少的设定，发展并分析更广义的核基插值方法。为此，我们首先在所选核函数与目标函数均满足最小化假设的条件下，给出广义插值的收敛性结果。最后，我们证明了针对全有界泛函集合的常用贪婪数据选择算法的收敛性。文中提供了支撑性数值结果以作说明。